Systematic generation of relativistic gaussian basis sets

A method for the Gaussian basis set generation for molecular relativistic Dirac-Fock calculations is proposed. The basis set exponents are obtained in the process of stochastic optimization (a hybrid of simplex and simulated annealing optimization techniques has been employed) of a functional defined as the sum of squares of differences between the numerical relativistic atomic wave functions and the wave functions obtained using the Gaussian function expansion. After this pre-optimization step the exponents are refined by ordinary gradient energy-functional based procedure. The present method seems to be very effective and robust. As an example the optimized basis sets of atoms from H (Z=1) to Ar (Z=18) are presented. Results of the Dirac-Fock calculations for all atoms under study are presented and compared with the numerical Dirac-Fock results and results obtained using the Gaussian basis sets according to Okada et al.: J. Chem. Phys.93 (1990) 5013. Slovak Grant Agency has funded the work reported in this paper, project No. 1/4205/98.     

Gauge-origin independent calculations of electric-field-induced second-harmonic generation circular intensity difference using London atomic orbitals

ABSTRACT

We present the first gauge-origin independent calculations of the circular intensity difference (CID) in electric-field-induced second-harmonic generation (EFISHG), including all contributions up to the electric-quadrupole–magnetic-dipole level. A recursive, open-ended response theory framework in combination with the use of London atomic orbitals allows us to ensure gauge-origin independent results. We apply this approach to study EFISHG-CID in a collection of chiral amino acids. We demonstrate that diffuse polarising basis functions are critical in order to obtain accurate CIDs, and that a basis set of at least aug-cc-pVTZ quality is needed in order to obtain results close to the basis-set limit. The use of London orbitals does not lead to significantly faster basis set convergence, although the improved basis set convergence allows the aug-cc-pVDZ basis set to be used with some confidence for larger molecules.     

Critical analysis of the modified firsov model. sensitivity to the choice of atomic wavefunctions

Abstract

The Firsov model for calculating the low-velocity e-lectronic stopping power (S_e) as modified by Cheshire, et al, is found to be extremely sensitive to the choice of atomic wave functions (ψ) when a minimum impact parameter is used in the calculations. Differences of a factor of ?2 are observed, for example, between the predicted value of S_e using a minimal basis set for ψ and the value obtained using an extended basis set. A systematic analysis to reduce the sources of uncertainty in calculations of S _e is proposed in order to have a unified set of criteria regarding the predictions of the theory.     

Numerical solution of the Sinanoǧlu equation using a multicentre radial-angular grid

ABSTRACT

The pair functions that minimise the correlation energy of second-order many-body perturbation (MP2) theory (the Hylleraas functional) are obtained as solutions to the corresponding Sinano?lu equation by expanding them on a six-dimensional, multicentre, radial-angular grid of two electrons. Cusps in the pair functions at the nuclei are described numerically accurately by the multicentre grid. A cusp in each singlet pair function at the coalescence of the two electrons is taken into account analytically by a correlation factor. With a grid of approximately 10,000 points per atom, the MP2 correlation energies for atoms and polyatomic molecules are obtained usually within 0.1 mE _h of the complete-basis-set results. The correlation factor, auxiliary basis functions, and a judicious choice of integration algorithms are all necessary to stabilise the grid-based MP2 and underlying Hartree–Fock (HF) calculations. The auxiliary basis set, in particular, largely restores the hermiticity and diagonal dominance of the Fock matrix as well as furnishes virtual orbitals used in a resolution-of-the-identity approximation to lower the dimension of some integrals. The results of the grid-based HF and MP2 calculations without a correlation factor are found to suffer from large, nonsystematic errors frequently.     

C 78 IPR fullerenes: Computed B3LYP/6-31G*//HF/3-21G temperature-dependent relative concentrations

The five isolated-pentagon-rule (IPR) satisfying isomers of C₇₈, labeled 1-5, or according to symmetry as D₃, C_2v, C' _2v , D_3h, and D' _3h , are computed. The cage geometries are optimized at the ab initio HF level with the standard 3-21G basis set (HF/3-21G). The separation energetics is then computed using the B3LYP density-functional treatment in the standard 6-31G* basis set (B3LYP/6-31G*//HF/3-21G). Harmonic vibrational frequencies are calculated by the SAM1 semiempirical method. The computed energies, structural and vibrational data are employed in the construction of isomeric partition functions and evaluation of the relative Gibbs free energies. The results are converted into relative concentrations for a wide temperature interval. The C' _2v structure is the most populated throughout while the D_3h species is negligible at all temperatures. The agreement between theory and experiment is reasonable, though some aspects are still to be clarified. Received 28 November 2000     

Application of the Truhlar basis set extrapolation procedure to ab initio calculations on van der Waals complexes

The simple basis set extrapolation procedure recently proposed by Truhlar [1998, Chem. Phys. Lett., 294, 45] has been applied to ab initio calculations on van der Waals clusters. Six test species, Ne₂, NeAr, Ar₂, ArH, ArFH and ArHF, were investigated. In general good agreement has been found with complete basis estimates of the van der Waals bond lengths and dissociation energies providing (i) the cc-pVDZ and cc-pVTZ basis sets originally employed by Truhlar are augmented by a single set of standard diffuse functions, and (ii) the counterpoise correction is used. Only ArHF yields disappointing results, presumably due to the substantial contribution of induction forces to the binding in this cluster. However, even in this case the calculated dissociation energy is within 15% of the complete basis set limit.     

A new basis set for molecular wavefunctions

A new basis set is proposed for molecular self-consistent field and configuration interaction calculations. Expansion functions are proposed in the form sin (r _A · p) exp (-ar_A ²) and cos (r _A · p) exp (-ar _A ²) and it is suggested that they are used as an alternative to the gaussian basis set x_A ⁱy_A ^jz_A ^k exp (-ar _A ²).

It is shown how the new basis can have the same effects, for suitable vectors p, as the gaussian basis. It is further shown that all the one and two-electron integrals in the new basis are evaluable in terms of exponentials, square roots and the complex error function erf (z). First calculations, using the new basis, are presented in LiH, HF and H₂O.     

New basis sets for the evaluation of the CO–Ne van der Waals complex interaction induced electric dipole moment and polarizability surfaces

Using the coupled cluster singles and doubles method together with the recently developed LPol-n (n?=?ds, dl, fs, fl), the aug-pc-2, the SVPD, the TZVPD and Dunning's x-aug-cc-pVXZ basis sets, we calculate the interaction induced electric dipole moment and polarizability of the CO–Ne van der Waals complex. We consider the effect of extending the bases with different sets of mid-bond functions, and after a systematic basis set study carried out at a representative set of intermolecular geometries, we select the aug-cc-pVTZ, the aug-pc-2, the LPol-fs and the TZVPD bases with a set of 3s3p2d1f1g mid-bond functions placed in the middle of the van der Waals bond for the evaluation of the whole interaction induced property surfaces. After having determined the optimal parameters of appropriate analytical functions fitting the interaction induced properties, the resulting surfaces are used in semiclassical calculations of the dielectric and refractivity second virial coefficients of the system. All through this study the results obtained with Dunning's basis set hierarchy and reported in Phys. Chem. Chem. Phys., 11, 9871 (2009) are taken as reference.     

Ab initio potential energy curve for the helium atom pair and thermophysical properties of dilute helium gas. I. Helium–helium interatomic potential

A helium–helium interatomic potential energy curve was determined from quantum-mechanical ab initio calculations. Very large atom-centred basis sets including a newly developed d-aug-cc-pV8Z basis set supplemented with bond functions and ab initio methods up to full CI were applied. The aug-cc-pV7Z basis set of Gdanitz (J. Chem. Phys. 113, 5145 (2000)) was modified to be more consistent with the aug-cc-pV5Z and aug-cc-pV6Z basis sets. The diagonal Born–Oppenheimer corrections as well as corrections for relativistic effects were also calculated. A new analytical representation of the interatomic potential energy was fitted to the ab initio calculated values. In a following paper this potential model will be used in the framework of quantum-statistical mechanics and of the corresponding kinetic theory to calculate the most important thermophysical properties of helium governed by two-body and three-body interactions.     

Optical Properties of Hexagonal Zink Sulfide in a Wide Energy Range of Fundamental Absorption

Four variants of full complexes of fundamental optical functions of hexagonal zinc sulfide are determined for E c and E c polarizations. The calculations are made on the basis of experimental reflectance and theoretical permittivity spectra using the Kramers–Kronig relations. The integral permittivity spectra are resolved into elementary components. The major parameters of each component (energy maxima, half-widths, areas, and oscillator forces of the transition bands) are determined. The key features of the spectra and transition component parameters are established. The data are interpreted on the basis of the well-known band calculations.     

Push it to the limit: comparing periodic and local approaches to density functional theory for intermolecular interactions

ABSTRACT

With the aim of systematically comparing two popular approaches to density functional theory – all-electron calculations with local basis sets, and periodic calculations employing plane wave basis sets and norm-conserving pseudopotentials – we have computed complete-basis binding energies across the S22 set of intermolecular interactions, a dataset consisting of noncovalent interactions of small- and medium-sized molecules containing first- and second-row atoms, using the Troullier-Martins norm-conserving pseudopotentials with SPW92, a local spin-density approximation; and PBE, a generalised gradient approximation. We have found that it is challenging to reach the basis set limit with these periodic calculations; for the methods and systems examined, a minimum vacuum distance of 30?Å between a system and its nearest images is necessary – unless some form of dipole correction is employed – as is a kinetic energy cutoff of at least 80 Ry. The trends in convergence with respect to vacuum size and kinetic energy cutoff are largely independent of the level of density functional approximation employed. A sense of the impact of each hyperparameter on basis set error provides a foundation for ensuring quality calculations in future studies and allows us to quantify the basis set errors incurred in existing studies on similar systems.     

First-principle calculation for the high-temperature diffusion coefficients of small pairs: the H–Ar Case

Argon has been widely used as a diluent for high-temperature reacting flow experiments. Numerical simulation of such process requires accurate diffusion coefficients for the H–Ar pair. All available potential energy functions for the H–Ar system have been empirically extrapolated in the repulsive region and are probably not reliable for prediction of H–Ar binary diffusion coefficient at high temperatures. We perform calculations using the restricted coupled cluster theory with single and double excitation (plus triple corrections) [RCCSD(T)] and suitable basis sets to obtain accurate potential energies. The calculated potential energy function is corrected for basis set superposition error and is validated against molecular beam scattering data. Comparisons with previous literature potential functions are also made. Using the Chapman–Enskog theory we carried out first-principle calculation of high-temperature diffusion coefficients by direct numerical integration of the collision integrals using the RCCSD(T) potential function. The computed diffusion coefficients are validated against available experimental data. Comparisons are also made with results obtained from transport compilations and packages commonly used in combustion simulation.     

On a Mathematical Framework for the Constitutive Equations of Anisotropic Dielectric Relaxation

Three classes of time-domain non-relativistic anisotropic dielectric constitutive equations of increasing generality are discussed. In each class dissipativity is ensured by the choice of a class of convolution kernels in the D-to-E constitutive equation expressing the electric field E in terms of the electric displacement field D. Defining properties of the inverse (E-to-D) kernels and their Fourier-Laplace transforms (complex dielectric functions) are determined by inversion of the D-to-E constitutive equation. By this procedure it is shown that dielectric functions of the dipolar dielectrics are tensor-valued Bernstein functions while the dielectric functions of the Drude-Lorentz type are tensor-valued negative definite functions. The properties of the complex dielectric permittivities are also determined for either class. The theory is applied to an exhaustive review of empirical response functions of real dielectric materials encountered in the literature. Each class of convolution kernels is consistent with existence of a conserved energy, but in one case a strictly dissipative energy can be constructed.     

A generalized internal axis method for high barrier tunneling problems,as applied to the water dimer

When more than one large-amplitude vibrational motion is present in a molecule, it is often not possible to define a global internal-axis-method (IAM) coordinate system and set of basis functions. In the present work, a method is presented for extending the IAM treatment to tunneling problems in such cases, using as an illustration a model for the water dimer with three large-amplitude vibrational coordinates. The method involves the construction of two different sets of local IAM-like coordinate systems. The first of these contains n coordinate systems, one for the small neighborhood surrounding each of the n equilibrium frameworks. The second contains on the order of $\text{n}{}^2\text{2}$ coordinate systems, one for each feasible tunneling path between each pair of frameworks. Basis functions written in the second set of local IAM-like coordinates are used to determine the complex phase factors associated in this method with tunneling matrix elements of the phenomenological rotational Hamiltonian in the high barrier limit. These phase factors govern the way in which the various real tunneling frequencies in the molecule constructively and/or destructively interfere in the Hamiltonian matrix elements and final energy expressions. Various mathematical approximations are involved in using the local IAM-like basis sets to obtain matrix elements; the full extent of the adverse effects of these approximations will not be known until an attempt to fit experimental data is carried out.     

Compact and accurate variational wave functions of three-electron atomic systems constructed from semi-exponential radial basis functions

The semi-exponential basis set of radial functions [A.M. Frolov, Phys. Lett. A 374, 2361 (2010)] is used for variational computations of bound states in three-electron atomic systems. It appears that the semi-exponential basis set has a substantially greater potential for accurate variational computations of bound states in three-electron atomic systems than was originally anticipated. In particular, the 40-term Larson’s wave function improved with the use of semi-exponential radial basis functions now produces the total energy –7.4780581457 a.u. for the ground 1²S-state in the ^￥Li^\infty{\rm Li} atom (only one spin function c₁\chi_1 = aba\alpha\beta\alpha - baa\beta\alpha\alpha was used in these calculations). This variational energy is very close to the exact ground state energy of the ^￥Li^\infty{\rm Li} atom and is substantially lower than the total energy obtained with the original Larson’s 40-term wave function (–7.477944869 a.u.).     

MRCISD calculations of the six lowest valence states of I2 −

Potential curves for the four Hund's case (a) basis electronic states (²Σ, ²Π_g, ²Π_u, and ²Σ⁺) correlating with the I(²P_u) + I^?(¹S_g) dissociation limit have been calculated at the MRCISD level of theory using an all-electron double-zeta basis set augmented with an extensive set of polarization and diffuse functions complete through i angular momentum functions. Transition moments for the dipole allowed transitions between these states are determined as a function of internuclear separation. The four Hund's case (a) curves are used to determine the six Hund's case (c) potential curves which arise when spin-orbit coupling is considered. The rovibrational eigenvalues for each of these six states are determined numerically, and standard spectroscopic constants are determined by fitting the energy of these levels to a Duham series. The results are compared with available experimental and theoretical information.     

Analyzing integrated-optical inhomogeneous planar waveguides by Galerkin's method: A detailed comparison of two different basis functions

The trigonometric and Hermite-Gaussian basis functions for determining the modal characteristics of inhomogeneous optical waveguides by means of the Galerkin's method are presented and analyzed. The results obtained with each set of basis functions for mode spectra and field distributions are compared with other exact and approximate methods. The merits and problems arising with each set of basis functions are discussed.